When will you be here?
NH has I think 18 miles of coast. You can drive right along it on I believe old route 1. It's a gorgeous drive, with lots of ocean scenery, and HUGE houses to look at. The warmest the water ever gets up to is about 68, and that's in late August. This time of year it's probably 40ish, maybe 50, but if you've never been in the Atlantic you might want to stick your feet in anway just to say you did (I'll be doing the same in the Pacific when we go to CA this summer).
The White Mountains are gorgeous. I believe it's actually a fairly straight shot from Rochester to the White Mountains, up route 16. There are some touristy places but there are also some scenic things. If you like water falls, Jackson Falls in Jackson and Glen Ellis Falls are both gorgeous. Great for picture taking. You might consider taking the gondola up to the top of Wildcat Mountain (it's a ski area in the winter). On a clear day you can see for 60 miles, and it's a great way to get up close and personal with Mt. Washington which is litterally right across the street.
Portsmouth is a neat little city on the water, not too far from Rochester. Lots of good places to eat, and some neat places to shop too. If you like history Strawberry Banke (in Portsmouth) is very historical. There's a park right across the street from there, Prescott Park, that's right on the river. If you're here in the summer, they have concerts there every weekend (free I believe).